Corollary 1

Tag 02J9, permanent. Cite it as [Bourbaki, Tag 02J9].

Théories spectrales, chapter II, § 1. Transformation de Fourier

Let

1\rightarrow H\overset{f}{\longrightarrow}G\overset{g}{\longrightarrow}K\rightarrow 1

be an exact sequence of commutative locally compact topological groups. The morphism $\widehat{g}$ induces an isomorphism between $\widehat{K}$ and $f(H)^{\bot}$ , and $\widehat{f}$ induces by passing to the quotient an isomorphism between $\widehat{G}/f(H)^{\bot}$ and $\widehat{H}$ .

By Theorem 4, the sequence

1\rightarrow \widehat{K}\widehat{\longrightarrow}^g\widehat{G}\widehat{\longrightarrow}^f\widehat{H}\rightarrow 1 \tag{32}

is exact. The morphism $\widehat{g}$ therefore induces an isomorphism of $\widehat{K}$ onto Ker( $\widehat{f}$ ) $=f(H)^{\bot}$ (Lemma 2 of II, p. 205), and $\widehat{f}$ induces by passing to the quotient an isomorphism of $\widehat{G}/Ker(\widehat{f}) =\widehat{G}/f(H)^{\bot}$ onto $\widehat{H}($ loc. cit.).

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

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